Proof of a refinement of Blum's conjecture on hexagonal dungeons

TL;DR

This paper refines Blum's conjecture on the enumeration of tilings of hexagonal dungeons, providing a more detailed formula and applying it to new dungeon types.

Contribution

It introduces a 3-parameter refinement of Blum's conjecture and extends the enumeration to additional hexagonal dungeon configurations.

Findings

Validated the refined conjecture through combinatorial enumeration.

Extended tiling enumeration to new hexagonal dungeon variants.

Demonstrated the applicability of graphical condensation methods.

Abstract

Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a,2a,b,a,2a,b$ (for $b\geq2a$) equals $13^{2a^2}14^{\lfloor a^2/2\rfloor}$. Ciucu and the author of the present paper proved the conjecture by using Kuo's graphical condensation method. In this paper, we investigate a 3-parameter refinement of the conjecture and its application to enumeration of tilings of several new types of the hexagonal dungeons.